To find the solution to [tex]\(\frac{5x^3 - 22x^2 - 17x + 11}{x-5}\)[/tex], let's proceed with polynomial division:
1. Identify the Polynomials:
- The numerator is [tex]\(5x^3 - 22x^2 - 17x + 11\)[/tex].
- The denominator is [tex]\(x - 5\)[/tex].
2. Perform Polynomial Division:
- When dividing [tex]\(5x^3 - 22x^2 - 17x + 11\)[/tex] by [tex]\(x - 5\)[/tex], we start by dividing the leading term of the numerator by the leading term of the denominator.
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x\)[/tex] to get the first term of the quotient: [tex]\(5x^2\)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\(5x^2\)[/tex] by [tex]\(x - 5\)[/tex], which gives [tex]\(5x^3 - 25x^2\)[/tex].
- Subtract [tex]\(5x^3 - 25x^2\)[/tex] from [tex]\(5x^3 - 22x^2 - 17x + 11\)[/tex], resulting in [tex]\(3x^2 - 17x + 11\)[/tex].
4. Repeat the Process:
- Now, divide [tex]\(3x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(3x\)[/tex].
- Multiply [tex]\(3x\)[/tex] by [tex]\(x - 5\)[/tex], giving [tex]\(3x^2 - 15x\)[/tex].
- Subtract [tex]\(3x^2 - 15x\)[/tex] from [tex]\(3x^2 - 17x + 11\)[/tex], resulting in [tex]\(-2x + 11\)[/tex].
5. Continue the Division:
- Divide [tex]\(-2x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-2\)[/tex].
- Multiply [tex]\(-2\)[/tex] by [tex]\(x - 5\)[/tex], giving [tex]\(-2x + 10\)[/tex].
- Subtract [tex]\(-2x + 10\)[/tex] from [tex]\(-2x + 11\)[/tex], resulting in a remainder of [tex]\(1\)[/tex].
6. Construct the Quotient and Remainder:
- The polynomial quotient is [tex]\(5x^2 + 3x - 2\)[/tex].
- The remainder is [tex]\(1\)[/tex].
Therefore, the result can be written as:
[tex]\[ \frac{5x^3 - 22x^2 - 17x + 11}{x - 5} = 5x^2 + 3x - 2 + \frac{1}{x - 5} \][/tex]
So, the final answer is:
[tex]\[ 5x^2 + 3x - 2 + \frac{1}{x - 5} \][/tex]
